www.warwick.ac.uk AUTHOR: Jack William Daniel Skipper DEGREE: MSc TITLE: Besov Spaces and Parabolic Systems DATE OF DEPOSIT: ................................. I agree that this thesis shall be available in accordance with the regulations governing the University of Warwick theses. I agree that the summary of this thesis may be submitted for publication. I agree that the thesis may be photocopied (single copies for study purposes only). Theses with no restriction on photocopying will also be made available to the British Library for microfilming. The British Library may supply copies to individuals or libraries. subject to a statement from them that the copy is supplied for non-publishing purposes. All copies supplied by the British Library will carry the following statement: “Attention is drawn to the fact that the copyright of this thesis rests with its author. This copy of the thesis has been supplied on the condition that anyone who consults it is understood to recognise that its copyright rests with its author and that no quotation from the thesis and no information derived from it may be published without the author’s written consent.” AUTHOR’S SIGNATURE: ....................................................... USER’S DECLARATION 1. I undertake not to quote or make use of any information from this thesis without making acknowledgement to the author. 2. I further undertake to allow no-one else to use this thesis while it is in my care. DATE SIGNATURE ADDRESS .................................................................................. .................................................................................. .................................................................................. .................................................................................. ..................................................................................
I agree that this thesis shall be available in accordance with the regulationsgoverning the University of Warwick theses.
I agree that the summary of this thesis may be submitted for publication.I agree that the thesis may be photocopied (single copies for study purposes
only).Theses with no restriction on photocopying will also be made available to the British
Library for microfilming. The British Library may supply copies to individuals or libraries.subject to a statement from them that the copy is supplied for non-publishing purposes. Allcopies supplied by the British Library will carry the following statement:
“Attention is drawn to the fact that the copyright of this thesis rests withits author. This copy of the thesis has been supplied on the condition thatanyone who consults it is understood to recognise that its copyright rests withits author and that no quotation from the thesis and no information derivedfrom it may be published without the author’s written consent.”
3 |φj(ξ)| ≥ Cε > 0 if ξ ∈ Rj,ε where Rj,ε := (2− ε)−12j ≤ |ξ| ≤ (2− ε)2j
4 |Dβφj(ξ)| ≤ Cβ2−j|β| for every β
7
5 (Sometimes require)∑∞
j=−∞ φj(ξ) = 1 or∑∞
j=−∞ φj(ξ) = δ(x).
Let Φ have the properties;
1. Φ ∈ S
2. Φ(ξ) 6= 0 iff ξ ∈ K where K := |ξ| ≤ 1
3. |Φ(ξ)| ≥ Cε 6= 0 if ξ ∈ Kε where Kε = |ξ| ≤ 1− ε.
Then with any such set of functions φjj∈Z,Φ we can define the Littlewood-
Paley decomposition Sj ,∆jj∈Z analogously to before.
There are a few simplifications that can be made on this set of eight proper-
ties. Firstly, we see that if we have φ0 with the properties 1 to 3 then we can define
φj by φj(ξ) = φ(ξ/2j). Further assuming that φ(ξ) ≥ 0 then we get 5 by replacing
φj(ξ) by φj(ξ)/∑∞
j=−∞ φj(ξ). Then in this case we can set Φ(ξ) =∑−1
j=−∞ φj(ξ).
With this cut off using elements of S, we can define the action on any element of S ′
and look at Lp norms of these decomposed functions. We can now define a Besov
space.
Definition 3.1.1 (Littlewood-Paley Besov space) Let s ∈ R, 1 ≤ p ≤ ∞, 0 <
q ≤ ∞. Then we set
Bsp,q = Bs
p,q(Rd) =
f : f ∈ S ′ and ‖S0f‖Lp +
∞∑j=0
(2js‖∆jf‖Lp
)q 1q
<∞
.
(3.1.1)
With the norm of,
‖f‖Bsp,q = ‖S0f‖Lp +
∞∑j=0
(2js‖∆jf‖Lp
)q 1q
.
The definition can be shown to be independent of the choice φj∞−∞ by
showing that this definition is equivalent to the interpolation space definition (5.1.3)
given later.
We now wish to define a homogeneous definition of a Besov space. To achieve
a homogeneous norm we must only care about the highest derivative and not any
fractions beneath as these will stop the scaling from working. Thus in the definition,
to keep it a normed space and not a semi-normed space, we remove lower degree
polynomials that only give an overall function shape structure and not the local
structure.
8
Definition 3.1.2 (Littlewood-Paley Homogeneous Besov space) Let s ∈ R,
1 ≤ p ≤ ∞, 0 < q ≤ ∞, N(s, d, p) > s − dp or with q ≤ 1 N(s, d, p) ≥ s − d
p , PNpolynomials of degree N . Then we set
Bsp,q = Bs
p,q(Rd) =
f : f ∈ S/P ′N and
∞∑j=−∞
(2js‖∆jf‖Lp
)q 1q
<∞
. (3.1.2)
With the semi-norm of,
‖f‖Bsp,q =
∞∑j=−∞
(2js‖∆jf‖Lp
)q 1q
modulo polynomials of degree PN .
For the Homogeneous case, instead of taking f ∈ S/P ′N we can define distri-
butions vanishing at infinity. The space S ′0(Rn) is the space of distributions such that
limm→−∞ Smf = 0 in S ′. This definition removes the polynomials as supp p = 0for p a polynomial, and also because p is a constant of the delta distribution.
3.2 Littlewood-Paley Properties
The first and possibly the most useful lemmas are the Bernstein Lemmas.
Lemma 3.2.1 (Bernstein) For A an annulus and B a ball there exists a constant
C such that for k ∈ Z and 1 ≤ p, q ≤ ∞ with p ≤ q and for any function u ∈ Lp,we have
As we see these give bounds, dependent on the support in Fourier space, for
the derivatives of a function. Therefore we see that the Littlewood-Paley decom-
position creates these nicely supported functions where we can apply the Bernstein
lemmas. These are useful to understand further properties and also used in showing
the equivalences of different definitions of Besov spaces.
Proof λ is just dilation so can assume λ = 1. For the first implication, let φ ∈ Dwith value 1 over B. As we can write u(ξ) = φ(ξ)u(ξ) so we have u(x) = φ ? u(x)
and thus ∂αu(x) = ∂αφ ? u(x). We can then apply Young’s inequality to obtain
‖∂αφ ? u‖Lq ≤ ‖∂αφ‖Lr‖u‖Lp ,
9
where 1r = −1
p + 1q + 1. To continue we must find a bound on the ‖∂αφ‖Lr term and
then we have the inequality we want. Notice that the use of Young’s inequality is
valid since we have p ≤ q. First we can state that
‖∂αφ‖Lr ≤ ‖∂αφ‖L∞ + ‖∂αφ‖L1 .
The sup-norm bounds the function over any compact set. The L1 bound is a tail
bound as functions with tails in L1 must decay faster than in any other Lp space.
We then want to collect the terms under one L∞ bound. This is done by multiplying
the function under the L1 bound by |x|2d
|x|2d . Then we take the sup of ∂α|x|2d out for
a bound. This gives
≤ C‖(1 + | · |2
)d∂αφ‖L∞ .
We then use that the Fourier transform maps L∞ to L1 and thus,
≤ C‖ (Id−∆)d ((·)α φ) ‖L1 ≤ Ck+1.
We then use the condition that φ ∈ D to get the bound we want.
Again we need to take a ψ ∈ D(Rd \ 0) with value 1 in neighborhood of
A. This is similar to before but now as the point 0 is removed this works for the
annulus case and we get the upper bound in a similar argument to above.
For the lower bound we again write u(ξ) = ψ(ξ)u(ξ). We can then, for some
constants Cα, write this term as.
u(ξ) = ψ∑|α|=k
(iξ)αCα(−iξ)α
|ξ|2du(ξ) =
∑|α|=k
(iξ)αu(ξ)Cα(−iξ)α
|ξ|2dψ
Then we can apply the inverse Fourier transform to both sides and we obtain
u(x) =∑|α|=k
(∂)αu(ξ) ? ψα where ψα = CαF−1
((−iξ)α
|ξ|2dψ
).
We can then apply Young’s inequality to both sides. To complete the proof
we need to bound ‖ψα‖L1 . This is done similarly to before and so we get the lower
bound we want.
One problem that needs to be considered is what to do when dealing with a
product of two functions when we are taking the Littlewood-Paley decomposition.
The method to deal with this is discussed in Bae [2009] and Bahouri et al. [2011].
For this explanation we shall absorb the S0 term into the ∆0 term and thus
can write the Littlewood-Paley decomposition of the function as∑
j≥0 ∆ju. For a
10
u and v in S ′ we then have that their product can be written as a Littlewood-Paley
decomposition by
uv =∑j,k
∆ju∆kv
where here j, k ∈ N.
This formula is a double sum over the product of all the combinations of the
decomposition. As this is complicated to deal with let us consider splitting the sum
up into three parts considering an arbitrary k and N0 dependent on the functions
used for the decomposition,
uv =∑j,k
∆ju∆kv =∑
k≤j−N0−1
∆ju∆kv +∑
k≥j+N0+1
∆ju∆kv +∑
|k−j|≤N0
∆ju∆kv.
As the support of the Fourier transform of each element is finite and the
Littlewood-Paley decomposition was chosen so that each support only intersects
with a finite number of the other terms in the decomposition, we can simplify these
three separate terms.
Looking at the first two terms, N0 is chosen so that the supports in Fourier
space no longer intersect. To make life easy with the specific example chosen earlier
we can choose N0 to be one. We can now simplify the first term by noticing that
as k is always less than j with independent support in Fourier space we can collect
the sum of these smaller terms into Sj−N0v. Similarly we can do this for the second
term.
We are now ready to define the paraproduct of v and u with the remainder
term as well as described in page nine of Danchin [2012].
Definition 3.2.2 (Paraproduct) The paraproduct of v by u is defined as.
Tuv :=∑j
Sj−N0u∆jv
The remainder of u and v is defined as.
R(u, v) :=∑
|j−ν|≤N0
∆ju∆νv
Thus with the paraproduct defined we can write the Bony decomposition of the
product of two functions uv .
Definition 3.2.3 (Bony decomposition) For the product of two functions uv
11
with u, v ∈ S ′ we can write the Bony decomposition as.
uv := Tuv + Tvu+R(u, v)
This gives a way to decompose products of functions. All this above can be done
for the homogeneous case as discussed in Bahouri et al. [2011] where the terms are
all the same yet the homogeneous Littlewood-Paley decomposition is used instead.
This decomposition will be used later in the proof of Onsager’s conjecture
where we have to deal with a term of products of functions to find a bound. Further
this is useful to look at embeddings of products of functions and there properties.
3.3 Littlewood-Paley Applications
Here we shall try to give some examples of applications where the Littlewood-Paley
definiton of a Besov space can be used to prove results for Besov spaces that other
spaces, such as Lp spaces or W sp spaces, do not necessarily satisfies.
3.4 Embeddings
One case of this is the embedding of H1(R2) into BMO but not into L∞ which
would be useful in many theorems. However what can be shown is the embedding
of B12,1 into L∞ which works as B1
2,1 is a slightly smaller space than H1 “yet is very
similar in size as H1 = B12,2”.
Example 3.4.1 B12,1(R2) embeds into L∞(R2)
Proof We know that from equation (3.0.1) B0∞,1 continuously embeds into L∞ so
want to show the embedding into B0∞,1 then we are done. Thus for a u ∈ B0
∞,1 it
has norm ‖S0u‖L∞ +∑
j∈N(‖∆ju‖L∞). Then we can multiply by 2j
2jand this gives.
‖S0u‖L∞ +∑j∈N
2j
2j(‖∆ju‖L∞) = ‖S0u‖L∞ +
∑j∈N
2j(1
2j‖∆ju‖L∞)
Thus all that is left to prove is that is that ( 12j‖∆ju‖L∞) ≤ ‖∆ju‖L2). This
is done by the scaling part of the Bernstein lemma (3.2.1) which gives ‖∆j‖Lq ≤
λd(
1p− 1q
)‖∆j‖Lp . Thus with λ = 2j , d = 2, p = 2 and q = ∞ we get the term 2j
out which cancels with the other 2j to give∑
j∈N 2j‖∆ju‖L∞ which is bounded by
assumption.
12
To finish the proof we notice the same scaling applies to the term ‖S0u‖L∞yet here λ = 1 and thus we have ‖S0u‖L∞ ≤ ‖S0u‖L2 trivially.
This important example is a specific case of an embedding theorem, Propo-
sition 2.20 in Bahouri et al. [2011], which states:
Proposition 3.4.1 Let 1 ≤ p1 ≤ p2 ≤ ∞ and 1 ≤ r1 ≤ r2 ≤ ∞. Then for s ∈ R
Bsp1,r1 is continuously embedded in B
s−d(
1p1− 1p2
)p1,r1 .
Proof This proof is similar to the proof above using ‖∆ju‖Lq ≤ λd(
1p1− 1p2
)‖∆j‖Lp
with λ = 2j . Then we just use the fact that lr1 is continuously embedded in lr2 , and
we are done.
Further we find in Proposition 2.39 that in a homogeneous setting the pre-
vious embedding in fact embeds into the space C0 (continuous functions that decay
to 0 at infinity). We can show this by noting that S0 is dense in Bdp
p,1 where in our
case is B12,1 as d = p = 2.
We see that this gives an interesting relation of the interplay between the
regularity and the integrability of functions in Besov spaces with more generality
than other embedding theorems.
3.5 Onsager’s Conjecture
3.5.1 Original Idea
Onsager’s conjecture considers the incompressible Euler equations given by the sys-
tem described in Shvydkoy [2010]
∂tu = (∇u)u−∇p Momentum balance (3.5.1)
∇ · u = 0 Incompressibility condition. (3.5.2)
Here u is a divergence-free velocity field, p is the internal pressure and consider the
entire space say R2 or R3 so we can use Fourier tools.
Here we care about the energy conservation of the system above. For an
initial condition u0 we obtain∫Ω|u(t)|2dx =
∫Ω|u0|2dx for all t ≥ 0.
We notice above that this condition holds if, once multiplying by a test
function u and integrating over time, the RHS of the equation (3.5.1) becomes
13
0. Now for smooth solutions with regular enough domains the RHS is zero and
conservation of energy holds. Assuming that we can perform integration by parts
with u|∂Ω = 0 and use the incompressibility condition then this term becomes 0.
The question is what is the minimal regularity needed for this conservation of energy
to hold.
Consider for instance that u is a distribution where we take u weakly con-
tinuous in time and L2 in space u ∈ L∞(0, t, L2) then this gives zero for the second
term on the RHS of (3.5.1). Thus to answer the question of minimal regularity,
our attention falls to the first term on the RHS of (3.5.1). Thus we care about the
total energy flux Π,
Π :=
∫Ω
(∇u)u · udx.
Then using the identity ∇ · (u ⊗ u) = (∇u)u + (∇ · u)u from Gonzalez and Stuart
[2008],
Π :=
∫Ω∇ · (u⊗ u) · u− (∇ · u)u · udx.
The condition u ∈ L∞(0, t, L2) does not allow us to integrate by parts here
and obtain that Π = 0. Here u ⊗ u is defined componentwise as the elements
uiuj for i, j = 1, . . . , d. We have seen that after performing a Littlewood-Paley
decomposition on a function, due to Bernstein lemmas, the derivative seems to act
like multiplication by constants and thus we could deduce that
Π ∼=∫
Ω(|∇|
13u)3dx.
So naıvely if u has Holder continuity 13 then Π would make sense and any
better regularity would be sufficient for integration by parts and give Π = 0. This
is Onsager’s conjecture that:
• Every weak solution to the incompressible Euler equations with smoothness
h > 13 conserves energy.
• Conversely there exists a weak solution to the incompressible Euler equations
of smoothness exactly 13 that does not conserve energy.
Onsager’s original heuristic justification was based on laws of turbulence and
not the Littlewood-Paley decomposition.
14
3.5.2 Critical Spaces and B133,c0
To explain what we mean by critical spaces let h ∈ C0 be a scalar mollifier with
dilation hδ(x) = 1δnh(xδ ). Then uδ(t, x) = u(t, ·) ?hδ(x) and for u ∈ L∞(0, t, L2) this
is well defined. Further, for δ > 0 uδ, is a smooth approximation of u and as δ → 0
we return u.
We can thus multiply (3.5.1) by (uδ)δ and integrate over space and time to
obtain∫ t
0
∫Rd∂tu · (uδ)δ dx ds+
∫ t
0
∫Rd
(∇u)u · (uδ)δ dx ds =
∫ t
0
∫Rd−∇p · (uδ)δ dx ds.
We can push the second mollification onto the other terms and with this
smoothness integrate by parts so that using incompressibility the RHS term van-
ishes. For the far left term, we can use the identity ∂u · u = 12∂t(u
2) and integrate
over time. Finally for the remaining term we can use the identity ∇ · (u ⊗ u) =
(∇u)u+ (∇ · u)u and integrate by parts to obtain
1
2(‖uδ(t)‖2L2
− ‖uδ(0)‖2L2) =
∫ t
0
∫Rn
(u⊗ u)δ : ∇uδ dx ds (3.5.3)
where A : B = Trace(AB). The RHS is the energy flux through different orders
scales δ. We notice that we have three u terms on the RHS. Thus for the optimal
function space we would want some bound in terms of ‖u‖3X in some function space
X involving time and space. Now with scale analysis we have these three outcomes:
• We have a one dimensional integral over time (T ), this will scale at rate T .
• We have three velocity terms, if we fix an average velocity (U) this will scale
at rate U3.
• We have one d-dimensional integral and one differential which together scale
with length(L) at a rate Ld−1.
Overall, this gives the formula to define an Onsager critical space as having
the scaling
(dim‖ · ‖X)3 = TU3Ld−1.
Some of these spaces in three dimensions are L3
(0, t, L 9
2
)and L3
(0, t,H
56
)when in
two dimensions we have L3(0, t, L6). Yet of particular interest are L3
(0, t, B
d(3−p)+p3p
p,r
)and thus L3
(0, t, B
133,l
)for l ∈ [1,∞] are critical for any dimension. In fact, the
15
proof will be done in the space L3
(0, T, B
133,c0
)where this is defined as follows
Bsp,c0 :=
u ∈ Bs
p,∞ : limj→∞
2js‖∆ju‖Lp = 0
.
3.5.3 Proving Onsager’s Conjecture in B133,c0
To prove this conjecture in the Besov space B133,c0
we will need the definition of the
weak solution to the incompressible Euler equations. We will also need a lemma
from Shvydkoy [2010] that relates this definition to Littlewood-Paley theory.
Definition 3.5.1 (Weak solution to Euler equations) A weakly continuous vec-
tor field u from [0, T ] to L2 u ∈ Cw(0, T, L2), is a weak solution to the Euler equa-
tions with initial data in u0 ∈ L2 if for every compactly supported test function
φ ∈ C∞0 ([0, T ]× Rd) with ∇ · φ = 0 and for every t ∈ [0, T ] we have∫Rd×t
u · φ dx−∫Rd×0
u · φ dx−∫ t
0
∫Rdu · ∂sφ dx ds =
∫ t
0
∫Rd
(u⊗ u) : ∇φ dx ds
(3.5.4)
and ∇u(t) = 0 in the sense of distributions.
With the next lemma one can pass from the weak formulation of the equation
to the mollified equation or to the integral equations. Then with the mollified
equation we can write this as a partial sum of the Littlewood-Paley decomposition
of the function u and then check the sum converges to 0.
Lemma 3.5.2 Let u be a weak solution to the incompressible Euler equations. Then
for each fixed δ > 0, uδ : [0, T ]→W sq is absolutely continuous for all s > 0 and q ≥ 2
and moreover
∂tuδ = −∇ · (u⊗ u)δ −∇pδ. (3.5.5)
Furthermore, (3.5.4) is equivalent to the integral equation,
u(t) = u0 −∫ t
0[∇ · (u⊗ u) +∇p] ds
in the sense of distributions for all t ∈ [0, T ].
We will have to introduce some more notation. We know that for a Littlewood-
Paley decomposition Sj ,∆jj∈Z we get the identity operator by Id = S0+∑
j≥0 ∆j .
Let us instead introduce the partial sum and define u≤q := (S0 +∑q
j≥0 ∆j)(u). So
16
this is only a partial Littlewood-Paley decomposition missing out the parts of u with
Fourier modes of the order 2k for k ∈ Z, k > q. We see that from the definition of
a mollifier that the process of mollification smooths out the modes of order 1δ and
greater. Thus as δ → 0 it smooths out only the higher modes until δ = 0 and we get
the identity. Thus we see the relation, for δ there is a q where the partial sum up to
q is the same as the mollified uδ. Further we can denote uq = (∆q−1 + ∆q + ∆q+1)u.
Finally we can define the Littlewood-Paley energy flux through wave number
2q by
Π≤q = −∫Rd
(u⊗ u)≤q : ∇u≤q dx. (3.5.6)
From the lemma above we know that if u(t) is a weak solution to the in-
compressible Euler equations then it satisfies (3.5.5). Thus from the discussion
above we can then write this mollifier equation in terms of partial Littlewood-Paley
decompositions as follows
∂tu≤q = −∇ · (u⊗ u)≤q −∇p≤q.
To this equation, we can multiply by u≤q and integrate over time and space.
Due to the absolute continuity of u≤q, as it is only a partial sum, we can follow the
same procedures as for equation (3.5.3) and obtain the equation.
1
2(‖u≤q(t)‖2L2
− ‖u≤q(0)‖2L2) = −
∫ t
0Π≤q(s) ds (3.5.7)
Now we can define the following localization kernel K = limq→∞Kq which
we will use later to give an important bound on the flux for u ∈ B133,∞. As described
in Shvydkoy [2010], the important feature of this bound is that it features a strongly
decreasing tail which greatly penalises far off interactions giving a quasi-local bound
on Π≤q.
Kq =
2q23 if q ≤ 0
2−q43 if q > 0.
With the following lemma we can easily prove the result. This interesting
bound to prove in Besov spaces for the operator Π≤q will finish off this proof.
Lemma 3.5.3 The energy flux of a divergence free vector field u ∈ B133,∞ satisfies
the following estimate.
|Π≤q| ≤ C∑j≥1
Kq−j2j‖∆ju‖3L3
(3.5.8)
17
where C > 0 is an absolute constant.
Theorem 3.5.4 (Onsager in B133,∞) Every weak solution u to the incompressible
Euler equations on a time interval [0, T ] that satisfies
limq→∞
∫ T
02q‖uq(t)‖3L3
dt = 0,
conserves energy on the entire interval [0, T ]. In particular, energy is conserved for
every solution in the class L3
(0, T, B
133,c0
).
Proof Taking the modulus of both sides of (3.5.7) we get the following
1
2(‖u≤q(t)‖2L2
− ‖u≤q(0)‖2L2) ≤
∫ t
0|Π≤q(s)|ds ≤ lim sup
q→∞
∫ t
0|Π≤q(s)|ds.
Then we can apply the above lemma to obtain the inequality
≤ lim supq→∞
∫ t
0C∑j≥1
Kq−j2j‖∆ju‖3L3
ds ≤ C lim supq→∞
∫ t
02q‖∆qu‖3L3
ds = 0.
The last inequality holds as the sum of Kq is exponentially decreasing for all j except
j = q then the limit as q →∞ is bounded except for j = q and so this term remains.
Proof (Proof of 3.5.3) To prove this we need to split up (u ⊗ u)≤q which we can
see as a matrix of all the multiplications of the components of u and thus to split
up this double sum we need to use paradifferential calculus and obtain the Bony
decomposition of the product of the functions Bahouri et al. [2011].
This gives the following from Constantin et al. [1994] and Shvydkoy [2010]
(u⊗ u)≤q = r≤q(u, u)− u>q ⊗ u>q + u≤q ⊗ u≤q
where,
r≤q(u, u)(x) =
∫Rd
Φq(y)(u(x− u)− u(x))⊗ (u(x− u)− u(x)) dy.
Thus substituting into the definition of Π≤q gives the equation
We now need to bound each term separately so for the first (1) term on the RHS of
(3.5.9) we use Holders inequality to give,
(1)RHS ≤ ‖r≤q(u, u)‖L 23
‖∇u≤q‖L3
Where further as 0 ≤ Φq ≤ 1 we obtain,
‖r≤q(u, u)‖L 23
≤∫Rd|Φq(y)|‖u(· − y)− u(·)‖2L3
dy.
Now we want to bound ‖u(· − y) − u(·)‖2L3. First we multiply by |y|
|y| inside the
modulus and see that we have
|y|2∥∥∥∥u(· − y)− u(·)
|y|
∥∥∥∥2
L3
.
The inside term looks like a differential. Due to the multiplication by the function
|Φq(y)| we see that this differential term only gives weight to Fourier modes of order
less than or equal to q. We can split it into two sums one over ≤ q and one over
> q and using Bernstein’s lemma for the differential term we obtain
‖u(· − y)− u(·)‖2L3≤∑p≤q|y|222p‖∆pu‖2L3
+∑p>q
‖∆pu‖2L3.
We want a bound with the term(
2p3 ‖∆pu‖L3
)3thus we multiply, inside the sum,
the first term by 243 (q−p)
243 (q−p)
and the second term by 223 (q−p)
223 (q−p)
and collect terms to give,
= |y|22q43
∑p≤q
2−43
(q−p)(2p3 ‖∆pu‖L3)2 + 2−
23q∑p>q
223
(q−p)(2p3 ‖∆pu‖)2
L3.
We notice that these sums form a pattern of the K kernel introduced before and
thus we can simplify to,(|y|22q
43 + 2−
23q)∑
p
Kq−p(2p3 ‖∆pu‖L3)2.
19
Substituting this back and using Bernstein inequality on ‖∇u≤q‖L3 we get that
(1)RHS ≤∫Rd|Φq(y)|
(|y|22q
43 + 2−
23q)∑
p
Kq−p(2p3 ‖∆pu‖L3)2 dy
∑p≤q
22p‖∆pu‖2L3
12
=∑p
Kq−p(2p3 ‖∆pu‖L3)2
(∫Rd|Φq(y)||y|22q
43 dy + 2−
23q
)∑p≤q
2p43
(2p3 ‖∆pu‖L3
)2
12
.
Now we need to calculate the integral. As |Φq(y)| is supported only for |y| ≤ 2−q
we can pull the supremum out of the integral to obtain∫Rd|Φq(y)||y|22q
43 dy ≤ 2−2q2q
43
∫Rd|Φq(y)| dy ≤ C2−q
23 .
Substituting this back in we obtain
(1)RHS ≤∑p
Kq−p(2p3 ‖∆pu‖L3)2C2−q
23
∑p≤q
2p43
(2p3 ‖∆pu‖L3
)2
12
= C∑p
Kq−p(2p3 ‖∆pu‖L3)2
∑p≤q
2−(q−p) 43
(2p3 ‖∆pu‖L3
)2
12
≤ C
(∑p
Kq−p(2p3 ‖∆pu‖L3)2
) 32
.
Then bringing this power inside the sum as greater than one we get the final result
(1)RHS ≤ C∑p
Kq−p(2p3 ‖∆pu‖L3)3.
We now have to deal with the other two terms, these are simpler. For instance the
(2)RHS of (3.5.9), with use of Holder’s inequality, similarly to before, we obtain
‖u>q‖2L3‖∇u≤q‖L3 . Then we can bound these terms again using the same methods
as above. Thus we are done.
20
Chapter 4
The Heat Kernel
Further equivalent definitions for a Besov space can be defined with use of the
heat kernel. We will see that the heat kernel definition, in general, has a rather
complicated structure. Yet there are occasions when this definition is useful for
example considering problems when the domain is not the whole space. Further, it
is useful when dealing with a problem which one can write a solution in terms of
the heat kernel. For instance, this is found in Cannone et al. [2004].
4.1 Heat Kernel Definition
We have two definitions from Lemarie-Rieusset [2010] for Besov spaces associated
to the heat kernel, for the non-homogeneous and the homogeneous cases.
Definition 4.1.1 (Non-Homogeneous Heat Kernel) For s ∈ R, 1 ≤ q ≤ ∞,
t0 > 0, α ≥ 0 so that α > s and for f ∈ S ′(Rd). Then for all t > 0
Bsp,q =
f : et∆f ∈ Lp and
(∫ t0
0
(‖t−
s2 (−t∆)
α2 et∆f‖Lp
)q dt
t
) 1q
<∞
(4.1.1)
With the equivalent norms,
‖f‖Bsp,q = ‖et0∆f‖Lp +
(∫ t0
0
(‖t−
s2 (−t∆)
α2 et∆f‖Lp
)q dt
t
) 1q
.
Note this norm is similar to norms defining Lorentz/interpolation spaces which we
will look at later.
We notice that the parameter j ∈ Z seems to have been replaced by the
conditions parameter 0 < t < ∞ as the countable sum is replaced by the integral
21
weighted by 1t . This weighted integral and sum perform the same measurement
on the function. This will be more obvious when we prove the equivalence of the
Littlewood-Paley definition and the heat kernel definition later.
The definition of the homogeneous case which is defined in Lemarie-Rieusset
[2010] and Bahouri et al. [2011] is introduced below.
Definition 4.1.2 (Homogeneous Heat Kernel) Let s < 0, 1 ≤ q ≤ ∞, f ∈S ′(Rd). Then for all t > 0
Bsp,q ∩ S ′0 =
f : et∆f ∈ Lp and
(∫ ∞0
(‖t−
s2 et∆f‖Lp
)q dt
t
) 1q
<∞
(4.1.2)
With the equivalent norms,
‖f‖Bsp,q =
(∫ ∞0
(‖t−
s2 et∆f‖Lp
)q dt
t
) 1q
We work in S ′0(Rn) as we again want to work modulo polynomials to allow us to
have a norm and not a semi-norm. As we know supp p = 0 for p a polynomial we
have to work in a space with these elements are removed.
We will show the equivalences between the Littlewood-Paley definitions and
the heat kernel definitions later after seeing bounds on the heat kernel for functions
f with suppf in some annulus.
4.2 Heat Kernel Properties
A useful bound given below describes the action of the heat kernel on functions whose
support in Fourier space is on an annulus. This is very useful for applications to look
at solutions to the heat equation. Further for showing properties of Besov spaces
and the equivalences in definitions using Littlewood-Paley and the heat kernel. The
next lemma is stated in Bahouri et al. [2011] and though it is stated on an annulus
it can further be shown to hold on a ball around 0 though c a constant in the lemma
is 0.
Lemma 4.2.1 (Heat Kernel Bound) For any annulus A there exists positive
constants c and C such that for any p ∈ [1,∞] and any pair of positive real numbers
(t, λ) we have
Supp u ⊂ λA =⇒ ‖et∆u‖Lp ≤ Ce−ctλ2‖u‖Lp . (4.2.1)
22
Proof As we are dealing with a function supported on an annulus A we again need
a function φ ∈ D(Rd \ 0) which is identically 1 near A to use in the proof. We
will force all the calculations on to it and with its nicer properties calculations will
be easier. Again we can let λ = 1, as this is just a dilation.
We multiply by φ which does not change the action of the heat kernel on A
as it has value one. As the Fourier transform of the heat kernel is e−t|ξ|2
we obtain
et∆u = F−1(φ(ξ)e−t|ξ|
2u(ξ)
)=
1
(2π)d
∫Rdeix·ξφ(ξ)e−t|ξ|
2dξ ? u.
Applying Young’s inequality to both sides gives the desired result provided we show
for c, C real positive constants and for all t > 0∥∥∥∥ 1
(2π)d
∫Rdeix·ξφ(ξ)e−t|ξ|
2dξ
∥∥∥∥L1
:= ‖f(t, ·)‖L1≤ Ce−ct.
To bound f in L1, the idea is to pull out a 1
(1+|x|2)das this is bounded in L1 and
then try to eliminate the(1 + |x|2
)dwe created. So first we multiply by 1 in the
correct manner
f(t, x) =1
(1 + |x|2)d
∫Rd
(1 + |x|2
)deix·ξφ(ξ)e−t|ξ|
2dξ.
Now we use the properties of the Fourier transform and integration by parts as φ(ξ)
compactly supported and thus we obtain
1
(1 + |x|2)d
∫Rd
((Id−∆ξ)
d eix·ξ)φ(ξ)e−t|ξ|
2dξ =
=1
(1 + |x|2)d
∫Rdeix·ξ (Id−∆ξ)
d φ(ξ)e−t|ξ|2dξ.
We then have to use Faa di Bruno’s formula, an algebraic identity, which
gives in our case for some constants Cαβ
(Id−∆ξ)d φ(ξ)e−t|ξ|
2=
∑β≤α,|α|≤2d
Cαβ
(∂(α−β)φ(ξ)
)(∂βe−t|ξ|
2).
Now as the support of φ is in an annulus we can find a couple of bounds. With c the
square of the lower bound of the support of φ and C being a function of the upper
bound of the support of φ. Also we notice that as φ ∈ D,∣∣∂(α−β)φ
∣∣ is bounded.
Further each ∂e−t|ξ|2
will drop down a constant of t and ξ yet the ξ can be bounded
23
by C so we have ∣∣∣(∂(α−β)φ(ξ))(
∂βe−t|ξ|2)∣∣∣ ≤ C(1 + t)|β|e−t|ξ|
2.
Then using the lower bound c we obtain
≤ C(1 + t)|β|e−ct.
Thus
‖f(t, ·)‖L1≤∫Rd
∣∣∣∣∣ 1
(1 + |x|2)d
∫Rdeix·ξ (Id−∆ξ)
d φ(ξ)e−t|ξ|2dξ
∣∣∣∣∣ dx≤∫Rd
1
(1 + |x|2)d
∫Rdeix·ξ
∣∣∣(Id−∆ξ)d φ(ξ)e−t|ξ|
2∣∣∣ dξ dx
≤∑
β≤α,|α|≤2d
CαβC(1 + t)|β|e−ct∫Rd
1
(1 + |x|2)ddx
∫Rdeix·ξ dξ
≤ Ce−ct∫Rd
1
(1 + |x|2)ddx
≤ Ce−ct.
This gives us the bound we needed. If we are dealing with a ball rather than the
annulus then the lower bound c = 0.
There is a corollary in Bahouri et al. [2011] that describes the behaviour of
the solution to the heat equation for initial conditions whose Fourier transform is
supported in an annulus. This behaviour relates the integrability over time to the
initial condition.
Corollary 4.2.2 Let A be an annulus and λ a positive real number. Let initial
condition u0 and forcing f(t, x) have its Fourier transform supported in λA for all
t ∈ [0, T ]. Consider the solution u(t) to the heat equation
∂tu− ν∆u = 0 and u(0, ·) = u0(·).
Consider v a solution of forced heat equation
∂tv − ν∆v = f and v(0, ·) = 0.
Then there exist positive constant C depending only on A such that for 1 ≤ a ≤ b ≤
24
∞ and 1 ≤ p ≤ q ≤ ∞ we have
‖u‖Lq(0,T,Lb) ≤ C(νλ2)− 1q λn(
1a− 1b )‖u0‖La ,
‖v‖Lq(0,T,Lb) ≤ C(νλ2)−1+
(1p− 1q
)λn(
1a− 1b )‖f‖Lp(0,T,La).
Proof For a u(t) satisfying the heat equation we can use the heat kernel to to
write u(t) = eνt∆u0. Then we start with the definition of ‖u‖Lq(0,T,Lb) and use the
lemma above (4.2.1) this gives
(∫ T
0
(∥∥eνt∆u0
∥∥Lb
)qdt
) 1q
≤(∫ T
0
(Ce−ctνλ
2 ‖u0‖Lb)q
dt
) 1q
.
We can then use Bernstein’s Lemma to get a bound from La to Lb and rearrange
(∫ T
0
(Ce−ctνλ
2λd(
1a− 1b ) ‖u0‖La
)qdt
) 1q
≤ Cλd(1a− 1b ) ‖u0‖La
(∫ T
0e−ctνλ
2q dt
) 1q
.
We can then integrate out to obtain
C ‖u0‖La λd( 1a− 1b )(
1
cνλ2q
) 1q (
1− e−cTνλ2q) 1q.
Thus as cTνλ2q > 0 we have e−cTνλ2q ∈ (0, 1) and we can bound
(1− e−cTνλ2q
) 1q ≤
1. Then absorbing constants gives the desired result.
For the second part we similarly start with v(t) =∫ t
0 eν(t−τ)∆f(τ) dτ and
the proof follows in a similar fashion.
As mentioned earlier we want to show the equivalence between the Littlewood-
Paley definition of a Besov space and the heat kernel definition. So far with the heat
kernel bound and the Bernstein’s lemmas we now have enough to prove the equiv-
alence of these two definitions. The proof will have inspiration from Bahouri et al.
[2011] yet here the proof is only for the homogeneous case where s < 0 and we look
at the non-homogeneous case for all s ∈ R. This is further mentioned in Lemarie-
Rieusset [2010].
Theorem 4.2.3 (Littlewood-Paley and heat kernel equivalence) The Littlewood-
Paley definition of a Besov space and the heat kernel definition are equivalent.
25
Proof First we want to look at the heat kernel integrand and using the heat kernel
bound above and the Bernstein lemma for the operator Iα = (−∆)α2 we have
‖t−s2 ∆j(−t∆)
α2 et∆u‖Lp ≤ Ct
α−s2 2jαe−ct2
2j‖∆ju‖Lp .
Then by multiplication by 2js
2jswe can pull the lower term with the 2jα term and take
the remaining term to create a Littlewood-Paley Besov norm after being summed
like so
‖t−s2 (−t∆)
α2 et∆u‖Lp =
∑j∈Z‖t−
s2 ∆j(−t∆)
α2 et∆u‖Lp ≤
≤ C∑j∈Z
tα−s2 2j(α−s)e−ct2
2j2js‖∆ju‖Lp ≤ C‖u‖Bsp,q
∑j∈Z
tα−s2 2j(α−s)e−ct2
2jcq,j .
Where cq,j is a generic element of the unit ball in lq(Z). This last inequality comes
from using Holder’s inequality to bound the last two terms on the left hand side in
lq and get the Littlewood-Paley Besov norm from the 2js‖∆ju‖Lp term. We get left
with a term in lq′ , for q′ the conjugate of q. Then we can use the duality definition
for an element in an lr space. (For an element in lr we know∑
j xryr′ < ∞ for all
xr ∈ lr and yr′ any element in the unit ball of lr′ .) We use this definition, of an
element in lq′ , for the rest.
For the rest of this proof we need something to deal with the spare term and
we need a lemma in Bahouri et al. [2011] which helps bound this term as we chose
α− s > 0.
Lemma 4.2.4 For any positive m, we have the bound
supt>0
∑j∈Z
tm2 2mje−ct2
2j<∞. (4.2.2)
From this lemma we first notice that for the case with q = ∞ we are done and
have the first inequality after taking the supremum over t > 0 of both sides of our
previous bound.
We now consider the case of q <∞. So we want to bound
∫ t0
0‖t−
s2 (−t∆)
α2 et∆u‖qLp
dt
t≤ C‖u‖qBsp,q
∫ t0
0
∑j∈Z
tα−s2 2j(α−s)e−ct2
2jcq,j
q
dt
t.
(4.2.3)
26
We can split the term inside the integral in preparation to use Holder’s inequality
∑j∈Z
tα−s2 2j(α−s)e−ct2
2jcq,j =
∑j∈Z
((tα−s2 2j(α−s)e−ct2
2j) 1qcq,j
)(tα−s2 2j(α−s)e−ct2
2j) q−1
q.
We then use Holder’s inequality with the first bracket term to the power of q and
the second q′ = qq−1 . This then, after we cancel the powers gives,
≤ C‖u‖qBsp,q
∫ t0
0
∑j∈Z
tα−s2 2j(α−s)e−ct2
2j
q−1∑j∈Z
tα−s2 2j(α−s)e−ct2
2jcqq,j
dt
t.
We take the supremum over time of the first term out and can bound by the previous
lemma. We only have to worry about the second term in the integral and we can
use Fubini’s theorem for this. To get the bound, we need to calculate the following
integral and show it is finite. To simplify, let k = α−s2
∑j∈Z
cqq,j
∫ t0
0tk−12j2ke−ct2
2jdt ≤
∑j∈Z
cqq,j
∫ ∞0
tk−12j2ke−ct22jdt.
Then we observe by calculation that∫∞
0 tk−1e−ct22jdt = C
2j2kΓ(k). Thus we find
that the whole term becomes
≤∑j∈Z
Ccqq,jΓ(k) ≤ CΓ(k).
We have the first inequality.
To prove the other direction, we use the fact that the last bound is of the
form of a gamma distribution to derive a useful identity
∆ju = 1 ? ∆ju =1
Γ(k + 1)
∫ ∞0
xke−x dx ? ∆ju.
Then use the substitution x = t|ξ|2 and the Fubini’s theorem, we obtain,
1
Γ(k + 1)
∫ ∞0
tk|ξ|2(k+1)e−x|ξ|2dx ? ∆ju =
1
Γ(k + 1)
∫ ∞0
tk|ξ|2(k+1)e−t|ξ|2? ∆ju dt.
Finally we can apply the inverse Fourier transform to both sides and we have,
∆ju =1
Γ(k + 1)
∫ ∞0
tk(−∆)k+1et∆∆ju dt. (4.2.4)
27
Now let k = α−s2 and split up et∆u = e
t2
∆et2
∆u. This lets us use the heat kernel
bound without losing the heat kernel properties for bounding the integral. After
performing these two substitutions we can bound the Lp norm with the usual use
of the Bernstein’s Lemmas and heat kernel bound
‖∆ju‖Lp ≤ C∫ ∞
0tα−s2 2j(−s+2)e−ct2
2j‖∆j(−∆)α2 e
t2
∆u‖Lp dt
≤ C∫ ∞
0tα−s2 2j(−s+2)e−ct2
2j‖(−∆)α2 et∆u‖Lp dt.
Now we want to look at the two cases again q = ∞ and q < ∞. For the first case
we want to bound supj∈Z(2js‖∆ju‖Lp
). From the above we obtain,
supj∈Z
(2js‖∆ju‖Lp
)≤ C sup
j∈Z
(2js∫ ∞
0tα−s2 2j(−s+2)e−ct2
2j‖(−∆)α2 et∆u‖Lp dt
).
Then we want to take the supremum over t out of the integral so that we obtain the
heat kernel definition and want the rest of the terms to be bounded
C supj∈Z
(2js sup
t>0
(tα−s2 ‖(−∆)
α2 et∆u‖Lp
)∫ ∞0
2j(−s+2)e−ct22jdt
).
We now just have to calculate the integral and rearrange We discover that everything
cancels nicely,
C supj∈Z
(2js2−js sup
t>0
(tα−s2 ‖(−∆)
α2 et∆u‖Lp
)2j2(−c2−2je−ct2
2j∣∣∣∞0
))=
= C supt>0
(tα−s2 ‖(−∆)
α2 et∆u‖Lp
).
We now have to look at the case of q < ∞. This is again going to use similar
methods to the q <∞ case before. We have to deal with the inequality below
∑j∈Z
(2js‖∆ju‖Lp
)q ≤ C∑j∈Z
(2js∫ ∞
0tα−s2 2j(−s+2)e−ct2
2j‖(−∆)α2 et∆u‖Lp dt
)q.
We can split this up to prepare for Holder’s inequality, used in the same way as
before,
C∑j∈Z
2jsq(∫ ∞
0e−ct22j 1
q tα−s2 2j(−s+2)e
−ct22j q−1q ‖(−∆)
α2 et∆u‖Lp dt
)q.
28
Then we can apply Holder’s inequality,
≤ C∑j∈Z
2jsq(∫ ∞
0e−ct2
2jdt
)q−1(∫ ∞0
tqα−s2 2qj(−s+2)e−ct2
2j‖(−∆)α2 et∆u‖qLp dt
).
Calculation of the first integral gives
C∑j∈Z
2jsq2−2j(q−1)
(∫ ∞0
tqα−s2 2qj(−s+2)e−ct2
2j‖(−∆)α2 et∆u‖qLp dt
).
Now we can use Fubini’s theorem to switch the sum and integral and then we can
simplify, this gives,
= C
∫ ∞0
∑j∈Z
2j2tqα−s2 e−ct2
2j‖(−∆)α2 et∆u‖qLp dt.
Multiplying by tt and rearranging gives us the measure we need for the integral in
the heat kernel definition of Besov space,
= C
∫ ∞0
tqα−s2 ‖(−∆)
α2 et∆u‖qLp
∑j∈Z
(t2j2e−ct2
2j) dt
t.
Then taking the supremum of the sum out of the integral and bounding by the
previous lemma, this rearranges to
≤ C∫ ∞
0‖t−
s2 (−t∆)
α2 et∆u‖qLp
dt
t.
This is the bound we wanted and together we have both the lower and upper bounds
so the equivalence is proved.
We have shown that the Littlewood-Paley definition and the heat kernel
definition are equivalent. This proves that the Littlewood-Paley definition must
be independent of the choice of functions used to create the decomposition. This
is because the heat kernel definition and the equivalence to the Littlewood-Paley
definiton is independent of the choice of functions used.
4.3 Heat Equation Applications
This application of Besov spaces relates the heat equation on R2 and the relations
between the initial conditions and the solution some time t later.
For the Sobolev case we discover that for u0 ∈ L2 we can achieve a bound
29
to show that at time t ∈ (0, T ], u(t) is in L1(0, T,H1) for any T . This calculation
will be given below and other simpler more general forms can be found in Evans
[1998]. We take the equation, multiply by u, and integrate over space and time then
perform integration by parts on the gradient term and use the identity utu = 12ddt |u|
2.
Switching integral over space and derivative gives
‖u(T )‖2L2+ 2
∫ T
0‖∇u‖2L2
dt = ‖u(0)‖2L2.
This gives the desired bound for controling the gradient with the L2 norm of the
initial condition.
We were able to increase the regularity by a derivative for the Sobolev case
above yet now we will show that with a slightly smaller Besov space than L2 we can
achieve extra regularity of two derivatives.
Example 4.3.1 Let the initial condition to the heat equation on Rd be u(0) = u0 ∈B0
2,1, then the solution at time t ∈ (0, T ] u(t) is in L1(0, T, B22,1) for any T .
This example shows how easy it is to use Besov spaces and specifically the
Littlewood-Paley definition for this calculation.
We see from (3.1.1) that for a function f ∈ B2p,q that f ∈ S ′ as well which
will be useful as the heat kernel is in S and thus the integrals will make sense.
Further we see that the space norm is the sum of two separate norms and thus we
will consider each one separately.
Proof We can write for u ∈ S ′ u(t) = et∆u0 as this is just the action of the heat
kernel.
We then want to consider the Littlewood-Paley decomposition of u(t) so first
consider the decomposition into the annulus so φj ? u(t) which we can also write
∆ju(t) for j ∈ N. We can write this part of the norm of u(t) by
∫ T
0
∣∣∣∣∣∣∞∑j=0
2j2∥∥et∆∆ju0
∥∥L2
∣∣∣∣∣∣ dt.Clearly the modulus can be ignored as the term inside is always positive. We
can now get the next inequality by the using the heat kernel estimate for a function
whose Fourier transform is supported in an annulus (4.2.1)∫ T
0
∞∑j=0
2j2∥∥et∆∆ju0
∥∥L2
dt ≤∫ T
0
∞∑j=0
2j2Ce−ct22j ‖∆ju0‖L2
dt.
30
We can now use Fubini’s theorem to switch sum and integral and obtain
∞∑j=0
2j2 ‖∆ju0‖L2
∫ T
0Ce−ct2
2jdt ≤
∞∑j=0
2j2 ‖∆ju0‖L2
1
c22j
(1− e−cT22j
).
This simplifies to
∞∑j=0
‖∆ju0‖L2
1
c
(1− e−cT22j
).
We see that cT22j is always positive so (1− e−cT22j ) is always in the interval
(0, 1] and so u0 is in B02,1 so this is bounded.
The Φ0 ? u(t) term can also be written S0u(t). We just need to show that∫ T0 ‖e
t∆∆ju0‖L2 is bounded. We can bound this by C∫ T
0 e−ct20‖∆ju0‖L2 . As c = 0
we just get a bound of C∫ T
0 ‖∆ju0‖L2 and we are done.
We notice here that if instead of q = 1 we used q = 2 and were using as our
spaces u0 ∈ L2 and u(t) ∈ L1(0, T,H2) then this proof would fail. This is clear as
the cancellation would not be complete and we would be left with a 22j term in the
sum. This extra term comes from the square of 22j we get from the l2 norm.
4.4 Heat Kernel on Sub Domains
There exists interesting problems that have been solved on the whole space but not
on bounded domains and therefore defining Besov spaces on these bounded domains
may help in solving some of these problems.
We have seen from the calculations that when working with problems on
the whole domain, the Littlewood-Paley definition seems to be the easiest to work
with. However, as seen before there is no analogous definition of the Littlewood-
Paley decomposition that can be defined on a bounded domain due to the innate
dependence on the Fourier transform.
We need to generalise the heat kernel definition to the domain Ω. For this
generalisation, one problem is the choice of boundary data for the Heat equation.
The choice of boundary data should depend on the values on the boundary that
one wants for elements of the Besov space. Therefore, if we are considering spaces
analogous to Wmp,0 where we want the functions to go to zero on the boundary we
31
would want to consider the heat equation with zero on the boundary condition.
d
dtu = ∆u in Ω
u = 0 on ∂Ω
u0 = u(0, ·) = f(·).
Then we wish to consider the u that solves this set of equations with the initial
condition f .
Definition 4.4.1 (Non-Homogeneous Heat Kernel norm on domain) For u
the solution to the equation above with f the initial data. For s ∈ R, 1 ≤ q ≤ ∞,
t0 > 0, α ≥ 0 so that α > s, α even positive integer. Then for all t > 0
‖f‖Bsp,q,0 = ‖u‖Lp +
(∫ t0
0
(‖t−
s2 (−t∆)
α2 u‖Lp(Ω)
)q dt
t
) 1q
Then to define the space it makes sense to have the value of the function
at the boundary to be zero. Thus taking inspiration from the Sobolev case we can
define, Bsp,q,0(Ω) ≡ f ∈ C∞0 (Ω) : ‖f‖Bsp,q,0(Ω) <∞
Bsp,q,0(Ω).
32
Chapter 5
Lorentz and Interpolation
Spaces
5.1 Background Theory
To define a real interpolation definition of a Besov space and, to give an initial idea
of how these definitions are all equivalent, Lorentz and interpolation spaces will
be introduced They are discussed in Peetre [1976] and Chapter 7 of Adams and
Fournier [2003]. These interpolation spaces are a useful building block for the Besov
spaces and useful in proofs of equivalence and properties of the Besov spaces.
These interpolation spaces and the associated Besov space definitions are
of importance though again not necessary easy to do calculations with. However
this definition is useful as it is more abstract so it can give general properties from
interpolation theory and is easier to generalise to other domains.
Definition 5.1.1 (Lorentz Space) Take a measure space (Ω,B(Ω), µ). If 0 <
p, q ≤ ∞ we define Lp,q the Lorentz space the space of j-measurable functions such
that
‖f‖Lp,q =
(∫ ∞0
(t1p f∗(t)
)q dt
t
) 1q
,
‖f‖Lp,∞ = supt>0
(t1p f∗(t))
and by convention L∞,∞ = L∞, where f∗ is the decreasing rearrangement of |f |
f∗(t) = infs : j|f | > s ≤ t.
33
Also Lp,∞ is the weak Lebesgue space. Furthermore L′p,q ≈ Lp′,q′ if 1 < p < ∞,1 ≤q <∞ or p = q = 1 here 1
p + 1p′ = 1.
These spaces then lead on to interpolation spaces which can be used to
generalise and treat many families of function spaces with the same approach.
Definition 5.1.2 (Banach couple) For two Banach spaces B0 and B1 and a Haus-
droff topological vector space B with B0 and B1 continuously embedded in B then we
define B = B0, B1 the Banach couple.
We now want to define an F that for the Banach couple B gives a Banach
space F (B) that is continuously embedded in B. This F (B) in an interpolation
space. It acts on the couple to take certain combinations of the elements of each
couple and creates a space out of this.
Real interpolation spaces have two equivalent definitions involving the differ-
ing functionals J and K described in Adams and Fournier [2003],Lemarie-Rieusset
[2010] and Peetre [1976]. Though, in Adams and Fournier [2003] and Lemarie-
Rieusset [2010] the definitions are formed where we first split the function in the
interpolation space into a diadic series first and then apply the J or K functional
to the series and check it is bounded. Here the definition given will be from Peetre
[1976].
Definition 5.1.3 (Lions and Gagliardo Real interpolation spaces) For the
Banach situation we introduce two auxiliary functionals K and J. For 0 < t < ∞,
b ∈ B0 +B1 we let
K(t, b; B) = K(t, b;B0, B1) = infb=b0+b1
(‖b0‖B0 + t‖b1‖B1)
If 0 < t <∞, b ∈ B0 ∩B1 we let
J(t, b; B) = J(t, b, B0, B1) = max (‖b‖B0 , t‖b‖B1)
Then let 0 < θ < 1, 0 < q ≤ ∞. Then we define the interpolation space (B)θ,q by
b ∈ (B)θ,q = (B0, B1)θ,q
⇐⇒(∫ ∞
0
(K(t, b)
tθ
)q dt
t
) 1q
<∞
34
⇐⇒ there exists u = u(t) (0 < t <∞) such that
(∫ ∞0
(J(t, u(t))
tθ
)q dt
t
) 1q
<∞ and b =
∫ ∞0
u(t)dt
t
⇐⇒ u = umu j ∈ Z such that
∞∑j=−∞
J(2j , uj)
2jθ<∞ and b =
∞∑j=−∞
uj
Here (B)θ,q has norm
‖b‖(B)θ,q=
(∫ ∞0
(K(t, b)
tθ
)q dt
t
) 1q
≈ infu
(∫ ∞0
(J(t, u(t))
tθ
)q dt
t
) 1q
≈ infu
∞∑j=−∞
(J(2j , uj)
2jθ
)q 1q
This dyadic decomposition interpolation definition is like the Littlewood-
Paley decomposition of a function mentioned earlier. This gives more understanding
of the relations between interpolation and the definitions involving integrals and
using the Littlewood-Paley definition where the integral is replaced by this countable
sum. We can use this definition to perform interpolation on the Littlewood-Paley
definition for calculation purposes.
Theorem 5.1.4 (Equivalence) In the definition above, the different sub defini-
tions are equivalent.
Proof (Sketch) First we notice that the first two norms are of the form
(∫ ∞0
(V
tθ
)q dt
t
) 1q
for some V and thus we only have to worry about V for the equivalences to hold.
For b ∈ (B)θ,q with the K functional definition we have infb=b0+b1(‖b0‖B0
+t‖b1‖B1). Thus if we then define b0/1 =∫∞
0 u0/1(t) dtt we have
infb=∫∞0 u0(t) dt
t+∫∞0 u1(t) dt
t
(∥∥∥∥∫ ∞0
u0(t)dt
t
∥∥∥∥B0
+ t
∥∥∥∥∫ ∞0
u1(t)dt
t
∥∥∥∥B1
), (5.1.1)
35
and by simple rearrangement of integrals we achieve
infb=∫∞0 u0(t) dt
t+∫∞0 u1(t) dt
t
(∫ ∞0‖u0(t)‖B0
dt
t+ t
∫ ∞0‖u1(t)‖B1
dt
t
). (5.1.2)
We know that (5.1.2) is bounded it implies that ‖u0(t)‖B0 and ‖u1(t)‖B1 are
bounded so the functional J = infu(max(‖u0(t)‖B0 , t‖u1(t)‖B1)) is bounded by a
constant of the K functional. Further, if (5.1.1) is bounded it implies b =∫∞
0 u(t) dtt
for some u.
For the other bound it is easy to see from (5.1.2) that if b =∫∞
0 u(t) dtt exists
and if J = infu(max(‖u0(t)‖B0 , t‖u1(t)‖B1)) is bounded then twice the maximum
will bound a linear combination of u0/1 and thus the K functional is bounded by a
constant of the J functional.
The proof for discrete K and J functionals is in Lemarie-Rieusset [2010] and
Adams and Fournier [2003] . Were we split the intergral into dyadic parts and prove
bounds Bernstein like bounds for each dyadic part.
One reason to look at these interpolation spaces is that the Lorentz spaces are
a specific class of interpolation spaces with the correctly chosen functional. Notice
Lp,q = [L1, L∞] 1p,q
More importantly the definitions of Besov spaces are defined so that we see
the norms are of the form of the interpolation spaces. We can, in fact, define the
Besov space via an interpolation of Sobolev spaces and then use theorems on the
properties of interpolation spaces to understand the properties of Besov spaces. For
instance the duality and reiteration theorems.
From the equivalence between this definition and the others we again show
that the Littlewood-Paley definition is independent on the choice of function used
in the decomposition. Further using embedding theorems for interpolation spaces
we can straight away get embedding theorems for Besov spaces for instance from
the comparison theorem in Peetre [1976] tells us that Bsp,1 →W s
p → Bsp,∞. Further
from the duality theorems we attain (Bsp,q)′ ∼= B−sp′,q′ as one would expect.
5.1.1 Real Interpolation Definition
Definition 5.1.5 (Real Interpolation Besov space) From Peetre [1976]. For
real interpolation, s real. We have for s = (1− θ)s0 + θs1 (0 < θ < 1)
(W s0p ,W
s1p
)θ,q
= Bsp,q.
36
Further this can be written
Bsp,q =
f :
(∫ ∞0
(K(t, f)
ts
)q dt
t
) 1q
<∞
, (5.1.3)
With
K(t, f) = K(t, f ;W s0p ,W
s1p ) = inf
f=f0+f1
(‖f0‖W s0
p+ t‖f1‖W s1
p
).
Interestingly one sees that from the interpolation definition of a Besov space,
we have to interpolate across two Sobolev spaces in the same Lp space but with
different derivative exponent. A function in the new space is bounded by the com-
bination s0θ + (1− θs1) with different outer Lq(dtt ) norms.
Therefore consider a function in a Besov space. We can bound this function
by taking a weighted sum of the norms of Sobolev functions with differentiability
either side. By varying θ we get a different weighting of the norms varying convexly
with θ.
Further the q determines the integrability we want for this weighting between
the norms and the measure dtt acts as a limit of the sums of these weights as discussed
earlier. Overall we see that the combination of these three indices allows for a strong
control on what functions we have in a Besov space.
We can define the definition for homogeneous real interpolation Besov spaces
similarly to above but using the homogeneous definition of a Sobolev space as defined
earlier.
Definition 5.1.6 (Homogeneous Real Interpolation Besov space) Same as
definition above but replace the space W s0p and W s1
p with W s0p and W s1
p .
5.2 Interpolation on Sub Domains
Finally we see from the definition that we can generalise this definition to bounded
domains as we can use Sobolev spaces with integer derivative coefficients which we
can define on bounded domains and then interpolate between them. As mentioned
in Chapter 2 there are many ways to define a Sobolev space on a bounded domain
and this will give many ways to define the bounded domain Besov space.
5.2.1 Interpolation Bounded Domain Definition
This is taken from an analogy to the Sobolev case where we can use interpolation
to define Sobolev spaces for any m ∈ R+.
37
Definition 5.2.1 (Real Interpolation Besov space bounded domain) For
Xmp,i defined as one of Hm
p (Ω),Wmp (Ω),Wm
p,0(Ω) for i respectively 1, 2, 3, m positive
integer. We have for s = (1− θ)m0 + θm1 (0 < θ < 1),(Xm0p,i , X
m1p,i
)θ,q
= Bsp,q,i.
Further can be written,
Bsp,q,i =
f :
(∫ ∞0
(Ki(t, f)
ts
)q dt
t
) 1q
<∞
, (5.2.1)
with
Ki(t, f) = Ki(t, f ;Xm0p , Xm1
p ) = inff=f0+f1
(‖f0‖Xm0
p,i+ t‖f1‖Xm1
p,i
).
We see here that we get three different definitions of a Besov space for each
different Sobolev space we interpolate over. The most useful for applications will
probably be the third where we use Wmp,0 as this definition has functions going to
zero on the boundary. This is useful to have in most applications.
Also in Chapter 7 of Adams and Fournier [2003], it defines a definition of
the bounded domain Besov space as above but with m0 = 0 thus we interpolate
between an Lp space and the Sobolev space with derivative order m > s for integer
m.
The interpolation can be done with the K or J functionals and possibly of
more interest the discrete K or J functionals and thus form a norm with instead
of the outer integral we can get a countable sum. This norm would be similar
to the Littlewood-Paley definitions and we may be able to use this definition for
applications assuming we can prove useful bounds like the Bernstein lemmas.
38
Chapter 6
Differential Difference and
Taibleson Poisson Integral
Definition
6.1 Differential Difference Definition
We have seen many different definitions so far for Besov spaces each with their own
uses. Here we shall introduce some more different equivalent definitions. These
definitions are again useful as there is no explicit use of the Fourier transform, so
generalisations to different domains can be done.
These definitions we initially defined for small ranges of s ∈ I ⊂ R, where
s is the differentiability of the space. Then we modified and generalised to include
any s ∈ R yet this causes the definitions to get more complicated.
These definitions can be formulated into a form similar to the interpola-
tion space definition yet with a different functional replacing K. Further there are
properties similar to the Littlewood-Paley definition when looking at the Fourier
transform of the functional.
In Peetre [1976] the Besov spaces can be defined under a generalisation of
the operator τhf(x) = f(x+ h)− f(x) and s ∈ (0, 1). This is
Bsp,q =
f : f ∈ Lp and
(∫Rd
(‖τhf‖Lp|h|s
)qdh
|h|d
) 1q
<∞
39
and in the case q =∞, one has
sup‖τhf‖Lp|h|s
<∞,
with the obvious norm.
We can see that inside the integral we have the form of a differential quotient
yet instead of bounding this under a supremum norm to get some Holder spaces we
are looking at a more generalised Lp norm.
We can generalise this for larger s firstly by differentiating out and then
checking if the following derivative of the function is in the 0 < s < 1 case. For
example
Bsp,q =
f : f ∈W k
p and Dαf ∈ Bs−kp,q (|α| ≤ k)
or this can be generlised by defining the kth difference operator
τkhf(x) =
k∑j=0
(−1)k((
k
j
)f(x+ hj)
)
so have the space for 0 < s < k as
Bsp,q =
f : f ∈ Lp and
(∫Rd
(‖τkhf‖Lp|h|s
)qdh
|h|d
) 1q
<∞
.
This can be written in another way that will help us see the connection between
this definition and the interpolation spaces defined earlier.
Definition 6.1.1 (Difference Besov space) Where ej = (0, · · · , 1, ·, 0) the jth
basis vector. 0 < s < k
Bsp,q =
f : f ∈ Lp and
(∫ ∞0
(‖τktejf‖Lp
ts
)qdt
t
) 1q
<∞ j = (1, · · · , n)
(6.1.1)
and in the case q =∞ one has the obvious L∞ modification.
6.2 Differential Difference in Sub Domain
6.2.1 Finite difference Bounded Domains Norm
For this norm on bounded domains, we are going to generalise the finite difference
Besov norm that we defined in the previous section.
40
This generalisation has two quite natural options:
• Firstly we can just make sure the finite difference stays within the space. Thus
we give the finite difference a value of 0 if [x, x+hej 6∈ Ω] and then just restrict
our Lp norms in the definition to Ω.
• Secondly we can keep our finite difference as we have before but integrate in
Lp over Ωh were Ωh ⊂ Ω and Ωh = ∩[s]+1j x : x+ h ∈ Ω.
As both generalisations give the same result here we will concentrate on the
first method of generalisation since it seems more natural to keep integrating over
the same domain when varying h.
We can generalise the kth difference operator by defining,
τkh,Ωf(x) =
∑k
j=0(−1)k((
kj
)f(x+ hj)
)[x, x+ kh] ⊆ Ω
0 Otherwise.
Thus we have the norm for 0 < s < k as
‖ · ‖Bsp,q(Ω) = ‖f‖Lp +
(∫Rd
(‖τkh,Ωf‖Lp(Ω)
|h|s
)qdh
|h|d
) 1q
.
This then simplifies to:
Definition 6.2.1 (Difference Besov norm bounded domain) Where ej =
(0, · · · , 1, ·, 0) the jth basis vector. 0 < s < k, k integer and Ω ⊂ Rd.
‖ · ‖Bsp,q(Ω) = ‖f‖Lp(Ω) +
n∑j=1
(∫ ∞0
(‖τktej ,Ωf‖Lp(Ω)
ts
)qdt
t
) 1q
(6.2.1)
and in the case q =∞ one has the obvious L∞ modification.
This norm is now the finite difference norm but restricted to the domain Ω.
We can now define Besov spaces with use of this norm similar to the Sobolev case
before in Chapter 2 and thus get three different associated spaces for this norm.
This definition and similar definitions of finite differences are used in Frehse
and Kassmann [2006], Kaminski [2011] and Buch [2006].
6.3 Taibleson Poisson Definition
This definition is interesting as it is derived from considering u(x, t) the (tempered)
solution to the PDE problem below. This again links the solution of PDEs to Besov
41
spaces like the heat kernel does. Consider the wave equation
∂2u
∂t2= −∆u if t > 0,
u = f if t = 0,
with solution given by the Poisson integral of f
u(x, t) =
∫Rd
t
((x− y)2 + t2)n+12
f(y) dy.
Then similarly to before for 0 < s < 1 we can define
Bsp,q =
f : f ∈ Lp and
(∫ ∞0
(‖t∂u∂t ‖Lp
ts
)qdt
t
) 1q
<∞
and we can extend this definition like before.
Then for general positive real number s we obtain the following definition.
Definition 6.3.1 (Poisson Besov space) For 0 < s < k
Bsp,q =
f : f ∈ Lp and
(∫ ∞0
(‖tk ∂ku
∂tk‖Lp
ts
)qdt
t
) 1q
<∞
(6.3.1)
and in the case q =∞ one has the obvious L∞ modification.
We could do a similar method to above but instead consider v = v(x, t) the
solution to the equation∂2v
∂t2= (1−∆)v if t > 0,
v = f if t = 0
instead and we get a similar definition of a Besov space to the one above.
We now want to look at the previous two examples and try to determine a
pattern in the definitions and see how this could link into earlier definitions. As
discussed in Peetre [1976], we notice the outer integrals are of the same form as
the interpolation space norm but with different integrands. So we just have to
worry about the integrands. We notice that they are of the form of a translation
invariant operator acting on f and therefore can be written as φt ? f where φt is
a test function depending on t of the form φt(x) = 1tnφ
(φt
). In terms of Fourier
transforms φt(ξ) = φ(tξ). This links to the Littlewood-Paley decomposition test
42
functions. In the cases above, the Fourier transform of the integrand becomes
τtejf(ξ) = (eitξj − 1)f(ξ).
Further we have
t∂u
∂t(ξ) = t|ξ|e−t|ξ|f(ξ).
Thus we would want a generalised definition to be of the form
Bsp,q =
f :∑
finite no. φ
(∫ ∞0
(t−s‖φt ? f‖Lp
)q dt
t
) 1q
<∞
(6.3.2)
with possible restrictions on φ and s.
To have a look for restrictions on φ we will use the embedding Bsp,1 →W s
p →Bsp,∞ from interpolation spaces. We find that φ must vanish in a neighborhood of 0
and ∞. Thus we get the Tauberian character
tξ : t > 0 ∩ φ 6= 0 6= for each ξ 6= 0
or in a stronger form
supp φ = b−1 < |ξ| < b with b > 1.
Usually b = 2 is chosen and we need a term ‖Φ ? f‖Lp where Φ 6= 0 = |ξ| < 1.We notice that the φt for the previous two definitions also share this property of
vanishing at zero and infinity. We also notice that they have values such that |φt| ≤ 1
and this is a further condition in the Littlewood-Paley decomposition so we see the
links between these definitions and the previous.
With stronger regularity conditions imposed this leads to the Littlewood-
Paley definition as we take φj ∈ S as well. As with this extra condition the
Fourier transform will make sense for all f ∈ S ′.
43
Bibliography
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H Bae. On the Navier-Stokes Equations. PhD thesis, New York University, 2009.
H Bahouri, J-Y Chemin, and R Danchin. Fourier analysis and nonlinear partial
